Missouri Journal of Mathematical Sciences

Perturbation Analysis for the Drazin Inverse Under Stable Perturbation in Banach Space

Yifeng Xue and Guoliang Chen

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Abstract

Let $X$ be Banach space and let $T, \bar T=T+\delta T$ be bounded linear operators on $X$. Suppose that $T$ has the Drazin inverse $T^D$ and $\text{Ind} (T)=n$. In this paper, we show that if $\|\delta T\|$ is sufficiently small and $\text{Ran} (\bar T^n) \cap \text{Ker} (({T^D})^n) = \{0\}$, then $\bar T$ is Drazin invertible with $\text{Ind} (\bar T)\le n$. In this case, the expression of $\bar T^D$ is given and the upper bounds of $\|\bar T^D\|$ and $$\dfrac{\|\bar T^D-T^D\|}{\|T^D\|}$$ are established. If $\dim X<\infty$, replacing $\text{Ran} (\bar T^n) \cap \text{Ker} (({T^D})^n) = \{0\}$ by $\text{rank} (\bar T^n) = \text{rank} (T^n)$, we obtain the same perturbation results of the Drazin invertible matrix $T$ as in the case of $\dim X=\infty$.

Article information

Source
Missouri J. Math. Sci., Volume 19, Issue 2 (2007), 106-120.

Dates
First available in Project Euclid: 15 September 2011

Permanent link to this document
https://projecteuclid.org/euclid.mjms/1316092490

Digital Object Identifier
doi:10.35834/mjms/1316092490

Zentralblatt MATH identifier
1169.47004

Subjects
Primary: 47A05: General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
Secondary: 65F20: Overdetermined systems, pseudoinverses 65J1

Citation

Xue, Yifeng; Chen, Guoliang. Perturbation Analysis for the Drazin Inverse Under Stable Perturbation in Banach Space. Missouri J. Math. Sci. 19 (2007), no. 2, 106--120. doi:10.35834/mjms/1316092490. https://projecteuclid.org/euclid.mjms/1316092490


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